* 1759-Leonhard Euler solves the partial differential equation for the vibration of a rectangular drum

In mathematics, the Euler – Tricomi equation is a linear partial differential equation useful in the study of transonic flow.

Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions.

* 1739-Leonhard Euler solves the ordinary differential equation for a forced harmonic oscillator and notices the resonance phenomenon

In this instance, sometimes the term refers to the differential equations that the system satisfies ( e. g., Newton's second law or Euler – Lagrange equations ), and sometimes to the solutions to those equations.

The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).

This is simply the Euler method for integrating the differential equation:

Illustration of numerical integration for the differential equation Blue: the Euler method, green: the midpoint method, red: the exact solution, The step size is

In calculus of variations, the Euler – Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary.

* Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.

In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements ; and in 1753 he applied the method to his study of the motions of the moon.

( It should be noted that Euler and Lagrange applied this method to nonlinear differential equations and that, instead of varying the coefficients of linear combinations of solutions to homogeneous equations, they varied the constants of the unperturbed motions of the celestial bodies.

In mathematics, a Cauchy – Euler equation ( also known as the Euler – Cauchy equation, or simply Euler's equation ) is a linear homogeneous ordinary differential equation with variable coefficients.

* R. Bott, L. Tu Differential Forms in Algebraic Topology: a classic reference for differential topology, treating the link to Poincaré duality and the Euler class of Sphere bundles

It is remarkable that this rule replaces the fairly complicated function of all three Euler angles, time derivatives of Euler angles, and inertia moments ( characterizing the rigid rotor ) by a simple differential operator that does not depend on time or inertia moments and differentiates to one Euler angle only.

His teachers recognized his talents in mathematics, but by 15 years of age he had already learned all the material taught at the school, and he began to study differential calculus from the works of Euler and Lagrange.

The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively.

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

His correspondence with Euler ( who also knew the above equation ) shows that he didn't fully understand logarithms.

The Euler equations can be integrated along a streamline to get Bernoulli's equation.

Neither the Navier-Stokes equations nor the Euler equations lend themselves to exact analytic solutions ; usually engineers have to resort to numerical solutions to solve them, however Euler's equation can be solved by making further simplifying assumptions.

) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.

Brahmagupta ( 628 CE ) started the systematic study of indefinite quadratic equations — in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler.

Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell.

In classical field theory, one writes down a Lagrangian density,, involving a field, φ ( x, t ), and possibly its first derivatives (∂ φ /∂ t and ∇ φ ), and then applies a field-theoretic form of the Euler – Lagrange equation.

Writing coordinates ( t, x ) = ( x < sup > 0 </ sup >, x < sup > 1 </ sup >, x < sup > 2 </ sup >, x < sup > 3 </ sup >) = x < sup > μ </ sup >, this form of the Euler – Lagrange equation is

and then applies the Euler – Lagrange equation, one obtains the equation of motion

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:

He worked with Euler on elasticity and the development of the Euler-Bernoulli beam equation.

The Euler – Tricomi equation is used in the investigation of transonic flow.

Others such as the Euler – Tricomi equation have different types in different regions.

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

As a result of surface area minimization, a surface will assume the smoothest shape it can ( mathematical proof that " smooth " shapes minimize surface area relies on use of the Euler – Lagrange equation ).

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

Euler proved in 1744 that the catenary is the curve which, when rotated about the x-axis, gives the surface of minimum surface area ( the catenoid ) for the given bounding circles.

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

The Euler – Maclaurin formula is also used for detailed error analysis in numerical quadrature.

In this way we get a proof of the Euler – Maclaurin summation formula by mathematical induction, in which the induction step relies on integration by parts and on the identities for periodic Bernoulli functions.

This way one can obtain expressions for ƒ ( n ), n = 0, 1, 2, ..., N, and adding them up gives the Euler – MacLaurin formula.

Sometimes is used, which is unfortunate since this is also used for the Euler characteristic

In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy.

For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for and the inverse tangent function.

It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Euler diagram for P ( complexity ) | P, NP, NP-complete, and NP-hard set of problems.

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.